Category Archives: Amplitudes Methods

Scattering Amplitudes at KITP

I’ve been visiting the Kavli Institute for Theoretical Physics in Santa Barbara for a program on scattering amplitudes. This week they’re having a conference, so I don’t have time to say very much.


The conference logo, on the other hand, seems to be saying quite a lot

We’ve had talks from a variety of corners of amplitudes, with major themes including the web of theories that can sort of be described by string theory-esque models, the amplituhedron, and theories you can “square” to get other theories. I’m excited about Zvi Bern’s talk at the end of the conference, which will describe the progress I talked about last week. There’s also been recent progress on understanding the amplituhedron, which I will likely post about in the near future.

We also got an early look at Whispers of String Theory, a cute short documentary filmed at the IGST conference.

The Road to Seven-Loop Supergravity

There’s an obvious way to put together a theory of quantum gravity. And it doesn’t work.

Do the same thing you would with any other theory, and you get infinity. You get repeated infinities, an infinity of infinities. And while you could fix one or two infinities, fixing an infinite number requires giving up an infinity of possible predictions, so in the end your theory predicts nothing.

String theory fixes this with its own infinity, the infinite number of ways a string can vibrate. Because this infinity is organized and structured and well-understood, you’re left with a theory that is still at least capable of making predictions.

(Note that this is an independent question from whether string theory can make predictions for experiments in the real world. This is a much more “in-principle” statement: if we knew everything we might want to about physics, all the fields and particles and shapes of the extra dimensions, we could use string theory to make predictions. Even if we knew all of that, we still couldn’t make predictions from naive quantum gravity.)

Are there ways to fix the problem that don’t involve an infinity of vibrations? Or at least, to fix part of the problem?

That’s what Zvi Bern, John Joseph Carrasco, Henrik Johansson, and a growing cast of collaborators have been trying to find out.

They’re investigating N=8 supergravity, a theory that takes gravity and adds on a host of related particles. It’s one of the easiest theories to get from string theory, by curling up extra dimensions in a particularly simple way and ignoring higher-energy vibrations.

Bern, along with Lance Dixon and David Kosower, invented the generalized unitarity technique I talked about last week. Along with Carrasco and Johansson, he figured out another important trick: the idea that you can do calculations in gravity by squaring the appropriate part of calculations in Yang-Mills theory. For N=8 supergravity, the theory you need to square is my favorite theory, N=4 super Yang-Mills.

Using this, they started pushing forward, calculating approximations to greater and greater precision (more and more loops).

What they found, at each step, was that N=8 supergravity behaved better than expected. In fact, it behaved like N=4 super Yang-Mills.

N=4 super Yang-Mills is special, because in four dimensions (three space and one time, the dimensions we’re used to in daily life) there are no infinities to fix. In a world with more dimensions, though, you start getting infinities, and with more and more loops you need fewer and fewer dimensions to see them.

N=8 supergravity, unexpectedly, was giving infinities in the same dimensions that N=4 super Yang-Mills did (and no earlier). If it kept doing that, you might guess that it also had no infinities in four dimensions. You might wonder if, at least loop by loop, N=8 supergravity could be a way to fix quantum gravity without string theory.

Of course, you’d only really know if you could check in four dimensions.

If you want to check in four dimensions, though, you run into a problem. The fewer dimensions you’re looking at, the more loops you need before N=8 supergravity could possibly give infinity. In four dimensions, you need a forbidding seven loops of precision.

(To compare, the highest precision of things we’ve actually tested in the real world is four loops.)

Still, Bern, Carrasco, and Johansson were up to the challenge. Along with Lance Dixon, David Kosower, and Radu Roiban, they looked at three loops, calculating an interaction of four gravitons, and the pattern continued. Four loops, and it was still going strong.

At around this time, I had just started grad school. My first project was a cumbersome numerical calculation. To keep me motivated, my advisor mentioned that the work I was doing would be good preparation for a much grander project: the calculation of whether the four-graviton interaction in N=8 supergravity diverges at seven loops. All I’d have to do was wait for Bern and collaborators to get there.

I named this blog “4 gravitons and a grad student”, and hoped I would get a chance to contribute.

And then something unexpected happened. They got stuck at five loops.

The method they were using, generalized unitarity, is an ansatz-based method. You start with a guess, then refine it. As such, the method is ultimately only as good as your guess.

Their guesses, in general, were pretty good. The trick they were using, squaring N=4 to get N=8, requires a certain type of guess: one in which the pieces they square have similar relationships to the different types of charge in Yang-Mills theory. There’s still an infinite number of guesses that can obey this, so they applied more restrictions, expectations based on other calculations, to get something more manageable. This worked at three loops, and worked (with a little extra thought) at four loops.

But at five loops they were stuck. They couldn’t find anything, with their restrictions, that gave the correct answer when “cut up” by generalized unitarity. And while they could drop some restrictions, if they dropped too many they’d end up with far too general a guess, something that could take months of computer time to solve.

So they stopped.

They did quite a bit of interesting work in the meantime. They found more theories they could square to get gravity theories, of more and more unusual types. They calculated infinities in other theories, and found surprises there too, other cases where infinities didn’t show up when they were “supposed” to. But for some time, the N=8 supergravity calculation was stalled.

And in the meantime, I went off in another direction, which long-time readers of this blog already know about.

Recently, though, they’ve broken the stall.

What they realized is that the condition on their guess, that the parts they square be related like Yang-Mills charges, wasn’t entirely necessary. Instead, they could start with a “bad” guess, and modify it, using the failure of those relations to fill in the missing pieces.

It looks like this is going to work.

We’re all at an amplitudes program right now in Santa Barbara. Walking through the halls of the KITP, I overhear conversations about five loops. They’re paring things down, honing their code, getting rid of the last few bugs, and checking their results.

They’re almost there, and it’s exciting. It looks like finally things are moving again, like the train to seven loops has once again left the station.

Increasingly, they’re beginning to understand the absent infinities, to see that they really are due to something unexpected and new.

N=8 supergravity isn’t going to be the next theory of everything. (For one, you can’t get chiral fermions out of it.) But if it really has no infinities at any loop, that tells us something about what a theory of quantum gravity is allowed to be, about the minimum necessary to at least make sense on a loop-by-loop level.

And that, I think, is worth being excited about.

Generalized Unitarity: The Frankenstein Method for Amplitudes

This is going to be a bit more technical than my usual, but you were warned.

There are a few things you’ll need to know to understand this post.

First, you should know that when we calculate probabilities of things happening in particle physics, we can do it by drawing Feynman diagrams, pictures of particles traveling and interacting. These diagrams can have loops, and the particle in the loop can have any momentum, from zero on up to infinity: you have to add up all the possibilities to get whatever you’re trying to calculate.

Second, you should understand that the “particles” in these loops aren’t really particles. They’re “virtual particles”, better understood as disturbances in quantum fields. Matt Strassler has a very nice article about this. In particular, these “particles” don’t have to obey E=mc^2 (or rather, if we include kinetic energy, E^2=p^2 c^2+m^2 c^4, where p is the momentum).

You can imagine a space that the momentum and energy “live in”. It’s got three dimensions for the three directions momentum can have, and one more dimension for the energy. Virtual particles can live anywhere in this four-dimensional space, but real particles have to live on a “shell” of points that obey E^2=p^2 c^2+m^2 c^4. If you’ve heard physicists say “on-shell” or “off-shell”, they’re referring to whether a particle is virtual, a quantum mechanical disturbance (and thus lives anywhere in the space) or a real classical particle (living on this “shell”).

Third, you should appreciate that in quantum physics, in Scott Aaronson’s words, we put complex numbers in our ontologies. Often, quantum weirdness shows itself when we look at our calculations as functions of complex numbers.

Let’s say I’m calculating an amplitude with one loop, and I draw a diagram like this:


Unitarity is how particle physicists say “all probabilities have to add up to one”. Since we have complex numbers in our ontologies, this statement is more complicated than it looks. One thing it ends up implying is that if I calculate an amplitude from the one-loop diagram above, its imaginary part will be given by multiplying together two simpler amplitudes:


Here you can imagine that I took a pair of scissors and “cut” the diagram in two along the dashed line. Now that the diagram has been “cut”, the particles I cut through are no longer part of a loop, so they’re no longer virtual: they’re real, on-shell particles.

If I wanted, I could keep “cutting” the diagram, generalizing this implication of unitarity. (For those who know some complex analysis, this involves taking residues.) I could cut all of the lines in the loop, like this:


Now something interesting happens. Here I’ve forced all four of the particles in the loop to be “on-shell”, to obey E^2=p^2 c^2+m^2 c^4. Previously, the momentum and energy in the loop was entirely free, living in its four-dimensional space. Now, though, it must obey four equations. And for those who’ve seen some algebra, four independent equations and four unknowns gives us one solution. By cutting all of these particles, we’ve killed all of the freedom that the loop momentum had. Instead of the living, quantum amplitude we had, we’ve cut it up into a bunch of dead, classical parts.

Why do this?

Well, suppose we have a guess for what the full amplitude should be. We’ve still got some uncertainty in our guess: it’s an ansatz.

If we wanted to check our guess, to fix the uncertainty in our ansatz, we could compare it to the full amplitude. But then we’d have to calculate the full quantum amplitude, and that’s hard.

It’s a lot easier, though, to calculate those “dead” classical amplitudes.

That’s the method we call “generalized unitarity”. We stitch together these easier-to-calculate, “dead” amplitudes. Enough different stitching patterns, and we can fix all the uncertainty in our ansatz, ending up with a unique correct answer without ever doing the full quantum calculation. Like Frankenstein, from dead parts we’ve assembled a living thing.


It’s off-shell!

How well does this work?

That depends on how good the ansatz is. The ansatze for one loop are very well understood, and for two loops the community is getting there. For higher loops, you have to be either smart or lucky. I happen to know some people who are both, I’ll be talking about them next week.

Thoughts from the Winter School

There are two things I’d like to talk about this week.

First, as promised, I’ll talk about what I worked on at the PSI Winter School.

Freddy Cachazo and I study what are called scattering amplitudes. At first glance, these are probabilities that two subatomic particles scatter off each other, relevant for experiments like the Large Hadron Collider. In practice, though, they can calculate much more.

For example, let’s say you have two black holes circling each other, like the ones LIGO detected. Zoom out far enough, and you can think of each one as a particle. The two particle-black holes exchange gravitons, and those exchanges give rise to the force of gravity between them.


In the end, it’s all just particle physics.


Based on that, we can use our favorite scattering amplitudes to make predictions for gravitational wave telescopes like LIGO.

There’s a bit of weirdness to this story, though, because these amplitudes don’t line up with predictions in quite the way we’re used to. The way we calculate amplitudes involves drawing diagrams, and those diagrams have loops. Normally, each “loop” makes the amplitude more quantum-mechanical. Only the diagrams with no loops (“tree diagrams”) come from classical physics alone.

(Here “classical physics” just means “not quantum”: I’m calling general relativity “classical”.)

For this problem, we only care about classical physics: LIGO isn’t sensitive enough to see quantum effects. The weird thing is, despite that, we still need loops.

(Why? This is a story I haven’t figured out how to tell in a non-technical way. The technical explanation has to do with the fact that we’re calculating a potential, not an amplitude, so there’s a Fourier transformation, and keeping track of the dimensions entails tossing around some factors of Planck’s constant. But I feel like this still isn’t quite the full story.)

So if we want to make predictions for LIGO, we want to compute amplitudes with loops. And as amplitudeologists, we should be pretty good at that.

As it turns out, plenty of other people have already had that idea, but there’s still room for improvement.

Our time with the students at the Winter School was limited, so our goal was fairly modest. We wanted to understand those other peoples’ calculations, and perhaps to think about them in a slightly cleaner way. In particular, we wanted to understand why “loops” are really necessary, and whether there was some way of understanding what the “loops” were doing in a more purely classical picture.

At this point, we feel like we’ve got the beginning of an idea of what’s going on. Time will tell whether it works out, and I’ll update you guys when we have a more presentable picture.


Unfortunately, physics wasn’t the only thing I was thinking about last week, which brings me to my other topic.

This blog has a fairly strong policy against talking politics. This is for several reasons. Partly, it’s because politics simply isn’t my area of expertise. Partly, it’s because talking politics tends to lead to long arguments in which nobody manages to learn anything. Despite this, I’m about to talk politics.

Last week, citizens of Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen were barred from entering the US. This included not only new visa applicants, but also those who already have visas or green cards. The latter group includes long-term residents of the US, many of whom were detained in airports and threatened with deportation when their flights arrived shortly after the ban was announced. Among those was the president of the Graduate Student Organization at my former grad school.

A federal judge has blocked parts of the order, and the Department of Homeland Security has announced that there will be case-by-case exceptions. Still, plenty of people are stuck: either abroad if they didn’t get in in time, or in the US, afraid that if they leave they won’t be able to return.

Politics isn’t in my area of expertise. But…

I travel for work pretty often. I know how terrifying and arbitrary border enforcement can be. I know how it feels to risk thousands of dollars and months of planning because some consulate or border official is having a bad day.

I also know how essential travel is to doing science. When there’s only one expert in the world who does the sort of work you need, you can’t just find a local substitute.

And so for this, I don’t need to be an expert in politics. I don’t need a detailed case about the risks of terrorism. I already know what I need to, and I know that this is cruel.

And so I stand in solidarity with the people who were trapped in airports, and those still trapped abroad and trapped in the US. You have been treated cruelly, and you shouldn’t have been. Hopefully, that sort of message can transcend politics.


One final thing: I’m going to be a massive hypocrite and continue to ban political comments on this blog. If you want to talk to me about any of this (and you think one or both of us might actually learn something from the exchange) please contact me in private.

Hexagon Functions Meet the Amplituhedron: Thinking Positive

I finished a new paper recently, it’s up on arXiv now.

This time, we’re collaborating with Jaroslav Trnka, of Amplituhedron fame, to investigate connections between the Amplituhedron and our hexagon function approach.

The Amplituhedron is a way to think about scattering amplitudes in our favorite toy model theory, N=4 super Yang-Mills. Specifically, it describes amplitudes as the “volume” of some geometric space.

Here’s something you might expect: if something is a volume, it should be positive, right? You can’t have a negative amount of space. So you’d naturally guess that these scattering amplitudes, if they’re really the “volume” of something, should be positive.

“Volume” is in quotation marks there for a reason, though, because the real story is a bit more complicated. The Amplituhedron isn’t literally the volume of some space, there are a bunch of other mathematical steps between the geometric story of the Amplituhedron on the one end and the final amplitude on the other. If it was literally a volume, calculating it would be quite a bit easier: mathematicians have gotten very talented at calculating volumes. But if it was literally a volume, it would have to be positive.

What our paper demonstrates is that, in the right regions (selected by the structure of the Amplituhedron), the amplitudes we’ve calculated so far are in fact positive. That first, basic requirement for the amplitude to actually literally be a volume is satisfied.

Of course, this doesn’t prove anything. There’s still a lot of work to do to actually find the thing the amplitude is the volume of, and this isn’t even proof that such a thing exists. It’s another, small piece of evidence. But it’s a reassuring one, and it’s nice to begin to link our approach with the Amplituhedron folks.

This week was the 75th birthday of John Schwarz, one of the founders of string theory and a discoverer of N=4 super Yang-Mills. We’ve dedicated the paper to him. His influence on the field, like the amplitudes of N=4 themselves, has been consistently positive.

Four Gravitons in China

I’m in China this week, at the School and Workshop on Amplitudes in Beijing 2016.


It’s a little chilly this time of year, so the dragons have accessorized

A few years back, I mentioned that there didn’t seem to be many amplitudeologists in Asia. That’s changed quite a lot over just the last few years. Song He and Yu-tin Huang went from postdocs in the west to faculty positions in China and Taiwan, respectively, while Bo Feng’s group in China has expanded. As a consequence, there’s now a substantial community here. This is the third “Amplitudes in Asia” conference, with past years meeting in Hong Kong and Taipei.

The “school” part of the conference was last week. I wasn’t here, but the students here seem to have enjoyed it a lot. This week is the “workshop” part, and there have been talks on a variety of parts of amplitudes. Nima showed up on Wednesday and managed to talk for his usual impressively long amount of time, finishing with a public lecture about the future of physics. The talk was ostensibly about why China should build the next big collider, but for the most part it ended up as a more general talk about exciting open questions in high energy physics. The talks were recorded, so they should be online at some point.

Hexagon Functions IV: Steinmann Harder

It’s paper season! I’ve got another paper out this week, this one a continuation of the hexagon function story.

The story so far:

My collaborators and I have been calculating “six-particle” (two particles collide, four come out, or three collide, three come out…) scattering amplitudes (probabilities that particles scatter) in N=4 super Yang-Mills. We calculate them starting with an ansatz (a guess, basically) made up of a type of functions called hexagon functions: “hexagon” because they’re the right functions for six-particle scattering. We then narrow down our guess by bringing in other information: for example, if two particles are close to lining up, our answer needs to match the one calculated with something called the POPE, so we can throw out guesses that don’t match that. In the end, only one guess survives, and we can check that it’s the right answer.

So what’s new this time?

More loops:

In quantum field theory, most of our calculations are approximate, and we measure the precision in something called loops. The more loops, the closer we are to the exact result, and the more complicated the calculation becomes.

This time, we’re at five loops of precision. To give you an idea of how complicated that is: I store these functions in text files. We’ve got a new, more efficient notation for them. With that, the two-loop functions fit into files around 20KB. Three loops, 500KB. Four, 15MB. And five? 300MB.

So if you want to imagine five loops, think about something that needs to be stored in a 300MB text file.

More insight:

We started out having noticed some weird new symmetries of our old results, so we brought in Simon Caron-Huot, expert on weird new symmetries. He couldn’t figure out that one…but he did notice an entirely different symmetry, one that turned out to have been first noticed in the 60’s, called the Steinmann relations.

The core idea of the Steinmann relations goes back to the old method of calculating amplitudes, with Feynman diagrams. In Feynman diagrams, lines represent particles traveling from one part of the diagram to the other. In a simplified form, the Steinmann conditions are telling us that diagrams can’t take two mutually exclusive shapes at the same time. If three particles are going one way, they can’t also be going another way.


With the Steinmann relations, things suddenly became a whole lot easier. Calculations that we had taken months to do, Simon was now doing in a week. Finally we could narrow things down and get the full answer, and we could do it with clear, physics-based rules.

More bootstrap:

In physics, when we call something a “bootstrap” it’s in reference to the phrase “pull yourself up by your own boostraps”. That impossible task, lifting yourself  with no outside support, is essentially what we do when we “bootstrap”: we do a calculation with no external input, simply by applying general rules.

In the past, our hexagon function calculations always had some sort of external data. For the first time, with the Steinmann conditions, we don’t need that. Every constraint, everything we do to narrow down our guess, is either a general rule or comes out of our lower-loop results. We never need detailed information from anywhere else.

This is big, because it might allow us to avoid loops altogether. Normally, each loop is an approximation, narrowed down using similar approximations from others. If we don’t need the approximations from others, though, then we might not need any approximations at all. For this particular theory, for this toy model, we might be able to actually calculate scattering amplitudes exactly, for any strength of forces and any energy. Nobody’s been able to do that for this kind of theory before.

We’re already making progress. We’ve got some test cases, simpler quantities that we can understand with no approximations. We’re starting to understand the tools we need, the pieces of our bootstrap. We’ve got a real chance, now, of doing something really fundamentally new.

So keep watching this blog, keep your eyes on arXiv: big things are coming.